Homomorphism space/Functorial properties/Fact

${\displaystyle \varphi \colon U\longrightarrow V}$

from another vector space ${\displaystyle {}U}$ induces a linear mapping

${\displaystyle \operatorname {Hom} _{K}{\left(V,W\right)}\longrightarrow \operatorname {Hom} _{K}{\left(U,W\right)},f\longmapsto f\circ \varphi .}$

${\displaystyle \psi \colon W\longrightarrow T}$

to another vector space ${\displaystyle {}T}$ induces a linear mapping

${\displaystyle \operatorname {Hom} _{K}{\left(V,W\right)}\longrightarrow \operatorname {Hom} _{K}{\left(V,T\right)},f\longmapsto \psi \circ f.}$